TY - JOUR
T1 - Efficient Reconstruction Methods for Nonlinear Elliptic Cauchy Problems with Piecewise Constant Solutions
AU - Egger , Herbert
AU - Leitao , Antonio
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 729
EP - 749
PY - 2009
DA - 2009/01
SN - 1
DO - http://doi.org/10.4208/aamm.09-m09S03
UR - https://global-sci.org/intro/article_detail/aamm/8394.html
KW - Nonlinear Cauchy problems, Elliptic operators, Level-set methods.
AB - <p style="text-align: justify;">In this article, a level-set approach for solving nonlinear elliptic Cauchy problems with piecewise constant solutions is proposed, which 
allows the definition of a Tikhonov functional on a space of 
level-set functions. We provide convergence analysis for the 
Tikhonov approach, including stability and convergence results. Moreover, a numerical investigation of the proposed Tikhonov regularization
method is presented. 
Newton-type methods are used for the solution of the optimality systems, 
which can be interpreted as stabilized versions of algorithms in a previous
work and yield a substantial improvement in performance. The whole approach is focused on three dimensional models, better suited 
for real life applications.</p>